# On the transfinite symmetric strong diameter two property

## TL;DR

This paper explores transfinite extensions of the symmetric strong diameter two property in Banach spaces, analyzing their stability under various operations and characterizing spaces with these properties using cardinal functions.

## Contribution

It introduces and studies transfinite analogues of the symmetric strong diameter two property, including stability results and characterizations for spaces of the form C_0(X).

## Key findings

- Stability of transfinite properties under c_0, ℓ_∞ sums, and tensor products.
- Characterization of C_0(X) spaces with these properties via cardinal functions.
- Examples of Banach spaces that do or do not have these properties.

## Abstract

We study transfinite analogues of the symmetric strong diameter two property. We investigate the stability of these properties under $c_0$, $\ell_\infty$ sums and under projective tensor products. Moreover, we characterize Banach spaces of the form $C_0(X)$, where $X$ is a T$_4$ locally compact space, which posses these transfinite properties via cardinal functions over $X$. As an application, we are able to produce a variety of examples of Banach spaces which enjoy or fail these properties.